In the standard classification of topological defects, in a theory with vacuum manifold $\mathcal{M}$,

  • $\pi_0(\mathcal{M})$ corresponds to domain walls,
  • $\pi_1(\mathcal{M})$ corresponds to strings/vortices,
  • $\pi_2(\mathcal{M})$ corresponds to monopoles,
  • $\pi_3(\mathcal{M})$ corresponds to textures.

In the last case, the $S^3$ we are mapping from is compactified space.

The Standard Model is sometimes said to have no topological defects, but its vacuum manifold $\mathcal{M} = SU(2)_L \times U(1)_Y /U(1)_A$ is $S^3$, which allows nontrivial textures. (Note that these are not the same thing as what goes into the electroweak theta vacua. The textures I'm describing here depend on $\pi_3(\mathcal{M})$, while the existence of those depends on $\pi_3(SU(2)_L)$.)

However, I've never heard of textures playing any role in the dynamics of the Standard Model, or in cosmology. Are there textures in the Standard Model, and if so, how do they behave?


1 Answer 1


Textures do play a role in the standard model and there are some cosmological models which suggest that the Baryon number asymmetry is related to their dynamics.

Topological defects related to the third homotopy group of the vacuum manifold $M$ are named Skyrmions or textures. Skyrmion configurations correspond to the case when the kinetic term is non-linear as in the low energy effective theory of QCD. These are stable solution, and there are plenty of indications that they describe the low energy dynamics of Baryons.

Texture configurations, on the other hand, correspond to the cases when the kinetic term is quadratic, i.e., of the Klein-Gordon type. The Higgs dynamical terms of grand unified theories are assumed to be of this type. For example, in a Higgs-doublet models with Lagrangians of the form: $$\mathcal{L} = D^{\mu}\Phi^{\dagger} D^{\mu}\Phi -\frac{1}{4g^2}\text{Tr} (F^{\mu\nu} F_{\mu\nu}) + \frac{\lambda}{4}( \Phi^{\dagger} \Phi - \eta^2)^2$$ Please, see for example, Nils-Erik Bomark's thesis.

When the theory is ungauged, the texture solutions $\Phi(x,t)$ are unstable due to Derrick's theorem, which asserts that the texture shrinks until it vanishes. These texture solutions are referred to as global textures. Since the texture solution corresponds to a topological invariant, a non-smooth transition takes place in the shrinking process towards the constant field configuration: When the texture shrinks to a small enough size, the gradients become large that it leaves the vacuum manifold and returns to it in the trivial configuration. Turok describes a model in which, this process is used to explain the large-scale structure of the universe.

When, the Higgs model is gauged, Derrick's theorem is no longer valid. The naive identification of all vacuum solutions due to the gauge invariance: $$\Phi(x) = g(x) \Phi_0 \sim \Phi_0$$ is not true due to the existence of large gauge transformations which define different states.

In the gauged case, the topological invariant is the difference between two winding numbers, each of which is not invariant under large gauge transformations, please see Bomark's thesis. $$N_H = -\frac{1}{24\pi^2} \int \epsilon^{ijk} \text{Tr}(\partial_i\Phi^{\dagger} \partial_j\Phi \partial_k\Phi^{\dagger} \Phi) $$ $$N_{CS}= \frac{g^2}{16\pi^2} \int \epsilon^{ijk} \text{Tr}(F_{ij}A_k + \frac{2ig}{3}A_iA_jA_k) $$ The vacuum (minimal energy solutions) corresponds to equal winding numbers $$N_H = N_{CS},$$

however the transition between the vacuua entails the passage through an energy barrier, which is a kind of instanton event called sphaleron. The change of the Chern-Simons winding number, through the axial anomaly causes a baryon number imbalance: $$\delta B = 3 \delta N_{CS}$$ Turok and Zadrozny use this mechanism to explain the baryon number imbalance in the universe. They argue that due to the CP violation and the departure from thermal equilibrium, the Baryon number has the correct sign.


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